Is it possible to remotely control a mySQL database. For instance, can I control the database from my house on a server with a webhosting company somewhere in new york (without having to login to my control panel).

with phpmyadmin you can manage a mysql database.
It's not important where the database is located or where the php-scripts are located. both can be at the other end of the world. As long as phpmyadmin can contact the database it doesn't matter.